Can I Determine the General Term for this Sequence?

[Michael_Light]

Homework Statement

The general term is given by 3r+1+(-1)^r+1

So by substituting r=0,1,2,3... I get a sequence like this: 0, 5, 6, 11, 12, 17, 18...

It seems to form some pattern. So i wonder, can i deduce the general term with only the sequence? How?

Please kindly elaborate more if possible because i really keen to learn how.

Homework Equations

The Attempt at a Solution

 

[Mentallic]

There aren't really any set methods to determine these general formulae for a sequence. You just need practice at solving them, and then it'll become easier for you to see what can be done, just like when simplifying trigonometric expressions and such.

Notice every second term has a difference of 6, so this is where you would expect a 3r to appear. But obviously there is another pattern within this pattern of 3r, so why don't we take 3r out of the equation and see if it makes the rest easier to find.

So if we take 3r out of
0, 5, 6, 11, 12, 17, 18...

We get
0, 2, 0, 2, 0, 2...

Now I'm sure with a little thought and understanding of the sequence (-1)^r, you can deduce the general expression. If it helps, take 1 away from the expression, so you end up with

-1, 1, -1, 1, -1...